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Full text of "Mathematical And Physical Papers - Iii"

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positive, and can only vanish in one very particular case. It denotes the vis viva consumed, or twice the work lost in the system during the time dt, in consequence of internal friction. According to the very important theory of Mr Joule, which is founded on a set of most striking and satisfactory experiments, the work thus apparently lost is in fact converted into heat, at such a rate, that the work expressed by the descent of 772 Ibs through one foot, supplies the quantity of heat required to raise 1 Ib. of water through 1° of Fahrenheit's thermometer.
50. The triple integral containing /u can only vanish when the differential coefficients of u, vy w satisfy the five following equations,
du _ dv __ dw dx~dy~~ dz : dw
dv    dw dz    dy
dx    dz ~~
du    dv _ .
dy    dx "~
These equations give immediately the following expressions for the differentials of u, v, w, in which the co-ordinates alone are supposed to vary, the time being constant:
du = MX - c0'"dy + a>"dz \
dv = §dy-a>fdz   + a"dx I...............(138).
dw = §dz — to'dx + <£>dy  }
In these equations S, <o', a>", a/" are certain functions of which the forms are defined by the equations (138), but need not at present be considered. It follows from equations (138) that the motion of each element of the fluid within the surface S is compounded of a motion of translation, a motion of rotation, and a motion of dilatation alike in all directions. So far as regards the first two kinds of motion, the fluid element moves like a solid, and of course there is nothing to call internal friction into play. For the reasons stated in my former paper, I was led to assume that a motion of dilatation alike in all directions (which of course can only exist in the case of an elastic fluid) has no effect in causing the pressure to differ from the statical pressure corresponding to the actual density, that is, in occasioning a violation of the functional relation commonly supposed to exist between the pressure, density, and temperature. The reader will observe that